B-Anomaly Expectations in SU(15) Models

• B-anomaly expectations are theoretical predictions derived from rare B-meson decays where LFU violations indicate potential new physics.
• The framework highlights the role of scalar leptoquarks in the SU(15) adjoint, naturally explaining the observed deficits in RK and RK* ratios.
• It further predicts testable signatures such as lepton flavor violation, exotic biquarks, and bileptons in collider and flavor experiments.

B-anomaly expectations comprise the set of theoretical and phenomenological implications, signatures, and model-building consequences inferred from observed deviations in rare semileptonic B-meson decays—most notably lepton flavor universality (LFU) ratios such as $R_K$ and $R_{K^*}$, where experimental data indicate significant violation of LFU. The subject crystallizes around interpreting these discrepancies as hints for new physics (NP), with specific attention to the structure and origin of the effective operators, the nature of plausible mediators, and the patterns of flavor violation implied by experimental results.

1. Definition and Experimental Context

Experimental B-anomalies refer to persistent, statistically significant tensions between Standard Model (SM) predictions and measurements in rare B-meson decay channels. Principal among these are LFU ratios: $$R_K = \frac{\Gamma(B^0 \to K^0 \mu^+ \mu^-)}{\Gamma(B^0 \to K^0 e^+ e^-)} \,,\quad R_{K^*} = \frac{\Gamma(B^0 \to K^{*0} \mu^+ \mu^-)}{\Gamma(B^0 \to K^{*0} e^+ e^-)}\,,$$ which are theoretically clean (SM: $R_{K^{(*)}}^{\text{SM}} \approx 1.00 \pm 0.01$) but exhibit experimental deficits ($R_K = 0.84 \pm 0.04$, $R_{K^*} = 0.69 \pm 0.09$), robustly indicating LFU violation. These anomalies center the expectation that new physics—if real—operates in mediating semileptonic B decays, and does so with non-universal couplings to different lepton flavors.

2. The Role of the Leptoquark and Mechanism of LFU Violation

The most theoretically efficient extension of the SM to resolve B-anomalies is the introduction of a leptoquark—a bosonic particle that couples directly to a quark and a lepton. In the framework of SU(15) grand unification, strong constraints on group theory representations and particle content uniquely point to a scalar (spin-0) leptoquark residing in the adjoint (224-dimensional) representation:

• Quantum numbers: triplet under $SU(3)_c$, singlet under $SU(2)_L$, electric charge $Q=+\frac{2}{3}$, hypercharge $Y=4/3$.
• Coupling hierarchy: The scalar leptoquark's Yukawa couplings can naturally scale as lepton masses ($y_\mu/y_e \sim m_\mu/m_e$), thereby breaking LFU in a manner directly proportional to known mass hierarchies.

The SU(15) adjoint uniquely provides this leptoquark in the correct gauge quantum numbers and with the proper coupling structure to explain the observed deficits in $R_K$ and $R_{K^*}$. For a scalar leptoquark ($LQ$), gauge invariance and allowed SU(15) couplings enable large differences between $e$ and $\mu$ channels in B decay amplitudes, consistent with all observed patterns of anomalies.

3. Effective Field Theory, Operator Structure, and Phenomenology

The presence of the scalar leptoquark generates new four-fermion operators modifying the effective Hamiltonian for $b\to s \ell^+\ell^-$ transitions: $$\mathcal{H}_{\text{eff}}^{bs\ell\ell} \supset \frac{y_{b s} y_{\ell\ell}}{M_{LQ}^2}\,(\bar{s}_L \gamma_\mu b_L)(\bar{\ell}_L \gamma^\mu \ell_L)\,,$$ with the coupling matrices reflecting the mass-driven LFU violation. These operators reduce the branching ratio for channels involving electrons relative to muons, thereby lowering $R_K$ and $R_{K^*}$ in agreement with experiment.

Further, the model predicts:

• LFU violation in tauonic B decays: Enhanced rates for $B^+ \to K^+ \tau^+ \tau^-$, $B_s \to \tau^+ \tau^-$.
• Lepton flavor violation (LFV): Owing to nontrivial Yukawa structures, rare processes such as $\tau \to \mu \gamma$, $\tau \to \mu \mu \mu$, and $B_s \to \tau \mu$ can become accessible.

4. Exotic State Content: Biquarks and Bileptons

SU(15) unification naturally embeds, alongside leptoquarks:

• Biquarks: Color-triplet or sextet, baryon-number-carrying exotic fermions with $L=0$.
• Bileptons: Non-SM gauge bosons coupling directly to lepton pairs.

Biquarks may impact the interpretation of exotic hadrons, while bileptons provide ancillary collider signatures. These states reside in the same SU(15) adjoint as the leptoquark, reinforcing the theoretical coherence of the framework.

5. Theoretical and Experimental Consequences

A scalar leptoquark in the SU(15) adjoint representation yields a suite of correlated predictions:

• Key quantitative expectations:
  • LFU ratios: $R_K \approx 0.84$, $R_{K^*} \approx 0.69$, SM values $\approx 1.00$.
  • Yukawa hierarchy: $|y_\mu/y_e| \sim 200$.
• No proton decay: Baryon number is a well-defined quantum number for all relevant states in 224, sidestepping the classic GUT problem.
• Experimental tests:
  • Enhanced $B \to K^{(*)} \nu \bar{\nu}$ branching fractions.
  • Measurement of LFV (e.g., $\tau \to \mu \gamma$) at levels exceeding SM predictions.
  • Direct collider searches for leptoquark, bilepton, and biquark resonances (LHC, future colliders).

Additional "smoking gun" signatures include strong deviations in LFU ratios for channels involving $\tau$ leptons, and the appearance of entirely new decay channels forbidden in the SM.

Property	Value
Spin	0 (Scalar)
Color	3 under $SU(3)_c$
Weak isospin	1 (Singlet under $SU(2)_L$)
Hypercharge	$4/3$, so $Q = +2/3$
Baryon number	$+1/3$ or $-1/3$ (multiplet-dependent)
Lepton number	$+1$ or $-1$
Coupling	Distinct Yukawas to $e$, $\mu$, $\tau$
Origin	Adjoint Higgs scalar of SU(15)

6. Group-Theoretic and Structural Considerations

The SU(15) GUT context offers unique group-theoretical features:

• The scalar leptoquark relevant to B-anomalies appears twice in the adjoint (224) and not at all in the 15, 105, or 120—highlighting the representation's exclusivity.
• All low-energy couplings and selection rules reflect the embedding of SM quantum numbers in the larger SU(15) symmetry.
• Simultaneous prediction of leptoquarks, biquarks, and bileptons in a unified algebraic setting.

7. Outlook and Open Directions

The SU(15) model with a scalar adjoint leptoquark constitutes a group-theoretically minimal and phenomenologically robust explanation for the ensemble of observed B-anomalies, with clear quantitative expectations for LFU and LFV observables, protected proton stability, and ancillary exotic state phenomenology. Near-future experimental advancements in LFU ratio measurements, LFV searches, and direct production of predicted exotics will further probe the viability of this scenario.